Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14)(7,11,16,25)(8,18,20,28)(9,21,23,13)(12,22,27,19)(15,29,30,36)(17,33,32,26)(24,34,35,31);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,26)(14,25)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8,10,20)(6,12,14,27)(7,15,16,30)(9,22,23,19)(11,24,25,35)(13,28,21,18)(17,34,32,31)(26,36,33,29);(7,16)(11,25)(15,30)(17,32)(24,35)(26,33)(29,36)(31,34)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #24411, #25183, #25186, … |
| maximal normal subgroup of | #24411, #25183, #25186, … |
| maximal quotient of | #19402, #19403, #19404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1776);